Split-complex numbers revisited

A split-complex number is an algebraic expression of the form x + j y where x, y \in \mathbb{R} and j^2 = + 1 where j \not = \pm 1. We can write a split-complex number simply as a pair of real numbers (x,y). Adding two split-complex numbers when represented as pairs is done component-wise. Hence, given (a, b) and (c, d) addition is defined as

    \[(a,b) + (c,d) = (a+c, b+d).\]

However, the product of two split-complex numbers is not defined component-wise since

    \[(a,b) \cdot (c,d) = (ac + bd, ad + bc).\]

One nice property of split-complex numbers is that we can change their representation in order to make multiplication component-wise as well.

Let

    \begin{align*}\iota = \frac{1 + j}{2} \ \mbox{ and } \ \iota^{*} = \frac{1 - j}{2}.\end{align*}

Now each hyperbolic number z = a + j b can uniquely be written in terms of
{\iota, \iota^* } as follows.

    \[z = (a + b) \iota + (a - b) \iota^{*}\]

From here, the multiplication of two numbers \tilde{z_1} = a \iota + b \iota^{*} and \tilde{z_2} = c \iota + d\iota^{*} is now given component-wise by

    \[ \tilde{z_1} \tilde{z_2} = a c \iota + b d \iota^{*}\]

In this sense the transformed split-complex numbers algebraically behave just like two copies of real numbers since the algebraic operations are performed in each copy independently. More on this alternative representation of split-complex numbers in subsequent posts.

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